Concentration fluctuations in aquifer transport: A rigorous first-order solution and applications

Flow and transport take place in a formation of spatially variable conductivity K(x). The latter is modeled as a lognormal stationary random space function. With Y = ln K, the structure is characterized by the mean [Y], the variance sigma(Y)(2), the horizontal and vertical integral scales I-h and I-v. The fluid velocity field V(x), driven by a constant mean head gradient, has a constant mean U and a stationary two-point covariance. Transport of a conservative solute takes place by advection and by pore-scale dispersion (PSD), that is assumed to be characterized by the constant longitudinal and transverse dispersivities alpha(dL) and alpha(dT). The local solute concentration C(x, t), a random function of space and time, is characterized by its statistical moments. While the mean concentration [C] was investigated extensively in the past, the aim here is to determine the variance sigma(C)(2), a measure of concentration fluctuations. This is achieved in a Lagrangean framework, continuous limit of the particle-tracking procedure, by adopting a few approximations. The present study is a continuation of a previous one (Dagan, G., Fiori, A., 1997. The influence of pore-scale dispersion on concentration statistical moments in transport through heterogeneous aquifers. Water Resour. Res., 33, 1595-1606) and extends it as follows: (i) it is shown that the indepence of the advective component of a solute particle trajectory from the trajectory component associated with PSD, is a rigorous first-order approximation in sigma(Y)(2). This independence, that was conjectured in the work of Dagan and Fiori (Dagan, G., Fiori, A., 1997. The influence of pore-scale dispersion on concentration statistical moments in transport through heterogeneous aquifers. Water Resour. Res., 33, 1595-1606), simplifies considerably the solution; (ii) the covariance of two-particle trajectories, needed in order to evaluate sigma(C)(2), is rederived, correcting for an error in the previous work. The general results are applied to determining CVC = sigma(C)/[C] at the center of a small solute body, of initial size much smaller than I-h = I-v, as function of sigma(gamma)(2), t' = tU/I and Pe = UI/D-dT = I/alpha(dT). Though PSD reduces considerably CVC as compared with advective transport (Pe = infinity), its value is still quite large for time intervals of interest in applications. This finding is in agreement with the analysis of field data by Fitts (Fitts, C.R., 1996. Uncertainty in deterministic groundwater transport models due to the assumption of macrodispersive mixing: evidence from the Cape Cod (Massachussets, USA) and Borden (Ontario, Canada) tracer tests. J. Contam. Hydrol., 23, 69-84). (C) 2000 Elsevier Science B.V. All rights reserved.